SYSREL Sample "Crash": Problem Description

Several Events must happen for Failure - A Parallel System

Preface

This example demonstrates the reliability analysis of a parallel system by means of SYSREL. We also illustrate using built in functions of the Symbolic Processor and setting up the logical model (SYSREL-specific) in addition to the failure functions and the stochastic model (equal in COMREL and SYSREL).

The failure probability of a structural element, e.g. the pile of a highway bridge, under truck impact can be determined by making use of the following simplified model. The impact force, based on energy balance, is:

where:

	: initial velocity
k	: equivalent stiffness
m	: total mass
a	: deceleration
	: pay load factor
	: angle between collision course and track direction
d	: distance from the structural element to the road

The geometrical settings are illustrated in the figure below.

Input: The Stochastic Model

The names (character identifiers) of the basic random variables together with the distribution type and moments or parameters are defined in the stochastic model window exactly like in COMREL (see the COMREL example). See below, how the character identifiers are used in the failure functions in the Symbolic Processor.

Basic Variable	Name of Variable	Distribution	Mean Value	Standard Deviation	C.o.V.(%)
v₀ (m/s)	Speed0	lognormal	22	2.2	10
k (kN/m)	Stiffness	lognormal	300	60	20.0
m (t)	Mass	normal	10.0	5.5	50
(-)	kappa	beta	0.7	0.1	0.14
a (m/s²)	Decelera	lognormal	4	1.3	32.5
(°)	Phi	Rayleigh	10	5.21	52.1
F_c(kN)	Fc	lognormal	median xsi	0.10	delta
U	U	normal	0	1	-

xsi and delta are the two parameters of the lognormal distribution (parameter type input)
U is an auxiliary standard normal variable for event no.3, see below.

The following Constant Parameters are used:

The names (character identifiers) of the constants together with the values are also defined in the stochastic model window.

Constant	Name of Constant	Comment	Value
d (m)	Dist	distance, see Figure	4.5
n (-)	n	number of trucks per day	2500
lambda (-)	lambda	rate for leaving track	10-10
t (days)	t	reference time	36500
xsi of F_c(kN)	xsi	median of max. interaction force	2000
b (m)	b	width of truck, see Figure	2.5

Input: Formulation of the Failure Function and of the Logical Model

The mechanical and probabilistic models are taken from an analysis about vehicle impact by T.Vrouwenfelder. There are three events to consider:

The event of failure of the structural element given that the truck reaches the structural element (unconditional probability)
The event that the truck does not come to a stop before hitting the target (hit probability given occurrence)
The event that a truck leaves the road.

The probabilities of the events are as follows:

Unconditional probability:
Hit probability given occurrence:
Occurrence probability:

The third event is introduced to model the discrete occurrence event. It depends on the random hit angle.

Next: Handling

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